Ebook Vector mechanics for engineers (9th edition): Part 1

Beer
Johnston

VECTOR MECHANICS
FOR ENGINEERS
VECTOR MECHANICS FOR ENGINEERS

MD DALIM #999860 12/18/08 CYAN MAG YELO BLK

ISBN 978-0-07-352940-0
MHID 0-07-352940-0
Part of
ISBN 978-0-07-727555-6
MHID 0-07-727555-1

Ninth Edition

www.mhhe.com

BEER | JOHNSTON | MAZUREK | CORNWELL | EISENBERG
Ninth Edition


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NINTH EDITION

VECTOR MECHANICS
FOR ENGINEERS

Statics and Dynamics
Ferdinand P. Beer
Late of Lehigh University

E. Russell Johnston, Jr.
University of Connecticut

David F. Mazurek
U.S. Coast Guard Academy

Phillip J. Cornwell
Rose-Hulman Institute of Technology

Elliot R. Eisenberg
The Pennsylvania State University


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VECTOR MECHANICS FOR ENGINEERS: STATICS & DYNAMICS, NINTH EDITION
Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the
Americas, New York, NY 10020. Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
Previous editions © 2007, 2004, and 1997. No part of this publication may be reproduced or distributed in any
form or by any means, or stored in a database or retrieval system, without the prior written consent of The
McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or
transmission, or broadcast for distance learning.
Some ancillaries, including electronic and print components, may not be available to customers outside the
United States.

This book is printed on acid-free paper.
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The credits section for this book begins on page 1291 and is considered an extension of the copyright page.
Library of Congress Cataloging-in-Publication Data
Vector mechanics for engineers. Statics and dynamics / Ferdinand Beer . . . [et al.]. — 9th ed.
p. cm.
Includes index.
ISBN 978–0–07–352940–0 (combined vol. : hc : alk. paper) — ISBN 978–0–07–352923–3
(v. 1 — “Statics” : hc : alk. paper) — ISBN 978–0–07–724916–8 (v. 2 — “Dynamics” : hc : alk. paper)
1. Mechanics, Applied. 2. Vector analysis. 3. Statics. 4. Dynamics. I. Beer, Ferdinand Pierre, 1915–
TA350.B3552 2009

620.1905—dc22
2008047184
www.mhhe.com


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About the Authors
As publishers of the books by Ferd Beer and Russ Johnston we are
often asked how they happened to write their books together with
one of them at Lehigh and the other at the University of Connecticut.
The answer to this question is simple. Russ Johnston’s first teaching appointment was in the Department of Civil Engineering and
Mechanics at Lehigh University. There he met Ferd Beer, who had
joined that department two years earlier and was in charge of the
courses in mechanics.
Ferd was delighted to discover that the young man who had
been hired chiefly to teach graduate structural engineering courses
was not only willing but eager to help him reorganize the mechanics
courses. Both believed that these courses should be taught from a few
basic principles and that the various concepts involved would be best
understood and remembered by the students if they were presented to
them in a graphic way. Together they wrote lecture notes in statics and
dynamics, to which they later added problems they felt would appeal
to future engineers, and soon they produced the manuscript of the first
edition of Mechanics for Engineers that was published in June 1956.
The second edition of Mechanics for Engineers and the first
edition of Vector Mechanics for Engineers found Russ Johnston at
Worcester Polytechnic Institute and the next editions at the University

of Connecticut. In the meantime, both Ferd and Russ assumed administrative responsibilities in their departments, and both were involved
in research, consulting, and supervising graduate students—Ferd in
the area of stochastic processes and random vibrations and Russ in the
area of elastic stability and structural analysis and design. However,
their interest in improving the teaching of the basic mechanics courses
had not subsided, and they both taught sections of these courses as
they kept revising their texts and began writing the manuscript of the
first edition of their Mechanics of Materials text.
Their collaboration spanned more than half a century and many
successful revisions of all of their textbooks, and Ferd’s and Russ’s
contributions to engineering education have earned them a number
of honors and awards. They were presented with the Western Electric
Fund Award for excellence in the instruction of engineering students
by their respective regional sections of the American Society for Engineering Education, and they both received the Distinguished Educator Award from the Mechanics Division of the same society. Starting in
2001, the New Mechanics Educator Award of the Mechanics Division
has been named in honor of the Beer and Johnston author team.
Ferdinand P. Beer. Born in France and educated in France and
Switzerland, Ferd received an M.S. degree from the Sorbonne and an
Sc.D. degree in theoretical mechanics from the University of Geneva.
He came to the United States after serving in the French army during

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iv

About the Authors

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the early part of World War II and taught for four years at Williams
College in the Williams-MIT joint arts and engineering program. Following his service at Williams College, Ferd joined the faculty of
Lehigh University where he taught for thirty-seven years. He held
several positions, including University Distinguished Professor
and chairman of the Department of Mechanical Engineering and
Mechanics, and in 1995 Ferd was awarded an honorary Doctor of
Engineering degree by Lehigh University.
E. Russell Johnston, Jr. Born in Philadelphia, Russ holds a B.S. degree
in civil engineering from the University of Delaware and an Sc. D. degree
in the field of structural engineering from the Massachusetts Institute of
Technology. He taught at Lehigh University and Worcester Polytechnic
Institute before joining the faculty of the University of Connecticut where
he held the position of Chairman of the Civil Engineering Department
and taught for twenty-six years. In 1991 Russ received the Outstanding
Civil Engineer Award from the Connecticut Section of the American
Society of Civil Engineers.
David F. Mazurek. David holds a B.S. degree in ocean engineering
and an M.S. degree in civil engineering from the Florida Institute of
Technology and a Ph.D. degree in civil engineering from the University of Connecticut. He was employed by the Electric Boat Division of
General Dynamics Corporation and taught at Lafayette College prior
to joining the U.S. Coast Guard Academy, where he has been since
1990. He has served on the American Railway Engineering and Maintenance of Way Association’s Committee 15—Steel Structures for the
past eighteen years. His professional interests include bridge engineering, tall towers, structural forensics, and blast-resistant design.
Phillip J. Cornwell. Phil holds a B.S. degree in mechanical engineering from Texas Tech University and M.A. and Ph.D. degrees in
mechanical and aerospace engineering from Princeton University. He
is currently a professor of mechanical engineering at Rose-Hulman
Institute of Technology where he has taught since 1989. His present
interests include structural dynamics, structural health monitoring,

and undergraduate engineering education. Since 1995, Phil has spent
his summers working at Los Alamos National Laboratory where he
is a mentor in the Los Alamos Dynamics Summer School and does
research in the area of structural health monitoring. Phil received an
SAE Ralph R. Teetor Educational Award in 1992, the Dean’s Outstanding Scholar Award at Rose-Hulman in 2000, and the Board of
Trustees Outstanding Scholar Award at Rose-Hulman in 2001.
Elliot R. Eisenberg. Elliot holds a B.S. degree in engineering and an
M.E. degree, both from Cornell University. He has focused his scholarly activities on professional service and teaching, and he was recognized for this work in 1992 when the American Society of Mechanical
Engineers awarded him the Ben C. Sparks Medal for his contributions
to mechanical engineering and mechanical engineering technology
education and for service to the American Society for Engineering
Education. Elliot taught for thirty-two years, including twenty-nine
years at Penn State where he was recognized with awards for both
teaching and advising.


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Contents
Preface

xv

List of Symbols xxiii

1

Introduction

1.1
1.2
1.3
1.4
1.5
1.6

What Is Mechanics? 2
Fundamental Concepts and Principles 2
Systems of Units 5
Conversion from One System of Units to Another
Method of Problem Solution 11
Numerical Accuracy 13

2

Statics of Particles

2.1

Introduction

1

10

14

16

Forces in a Plane 16
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11

Force on a Particle. Resultant of Two Forces 16
Vectors 17
Addition of Vectors 18
Resultant of Several Concurrent Forces 20
Resolution of a Force into Components 21
Rectangular Components of a Force. Unit Vectors 27
Addition of Forces by Summing x and y Components 30
Equilibrium of a Particle 35
Newton’s First Law of Motion 36
Problems Involving the Equilibrium of a Particle.
Free-Body Diagrams 36

Forces in Space

45

2.12 Rectangular Components of a Force in Space 45

2.13 Force Defined by Its Magnitude and Two Points on Its
Line of Action 48
2.14 Addition of Concurrent Forces in Space 49
2.15 Equilibrium of a Particle in Space 57
Review and Summary 64
Review Problems 67
Computer Problems 70

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3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10

3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
*3.21

Rigid Bodies: Equivalent
Systems of Forces 72
Introduction 74
External and Internal Forces 74
Principle of Transmissibility. Equivalent Forces 75
Vector Product of Two Vectors 77
Vector Products Expressed in Terms of
Rectangular Components 79
Moment of a Force about a Point 81
Varignon’s Theorem 83
Rectangular Components of the
Moment of a Force 83
Scalar Product of Two Vectors 94
Mixed Triple Product of Three Vectors 96
Moment of a Force about a Given Axis 97
Moment of a Couple 108
Equivalent Couples 109
Addition of Couples 111

Couples Can Be Represented by Vectors 111
Resolution of a Given Force into a Force at O
and a Couple 112
Reduction of a System of Forces to One Force and
One Couple 123
Equivalent Systems of Forces 125
Equipollent Systems of Vectors 125
Further Reduction of a System of Forces 126
Reduction of a System of Forces to a Wrench 128

Review and Summary 146
Review Problems 151
Computer Problems 154

4

Equilibrium of Rigid Bodies

4.1
4.2

Introduction 158
Free-Body Diagram 159

156

Equilibrium in Two Dimensions 160
4.3
4.4
4.5

4.6
4.7

Reactions at Supports and Connections
for a Two-Dimensional Structure 160
Equilibrium of a Rigid Body in Two Dimensions
Statically Indeterminate Reactions. Partial
Constraints 164
Equilibrium of a Two-Force Body 181
Equilibrium of a Three-Force Body 182

162


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Contents

Equilibrium in Three Dimensions 189
4.8
4.9

Equilibrium of a Rigid Body in Three Dimensions
Reactions at Supports and Connections for a
Three-Dimensional Structure 189

189

Review and Summary 210
Review Problems 213
Computer Problems 216

5
5.1

Distributed Forces: Centroids
and Centers of Gravity 218
Introduction

220

Areas and Lines
5.2
5.3
5.4
5.5
5.6
5.7
*5.8
*5.9

220

Center of Gravity of a Two-Dimensional Body 220
Centroids of Areas and Lines 222
First Moments of Areas and Lines 223
Composite Plates and Wires 226
Determination of Centroids by Integration 236

Theorems of Pappus-Guldinus 238
Distributed Loads on Beams 248
Forces on Submerged Surfaces 249

Volumes

258

5.10 Center of Gravity of a Three-Dimensional Body.
Centroid of a Volume 258
5.11 Composite Bodies 261
5.12 Determination of Centroids of Volumes by
Integration 261
Review and Summary 274
Review Problems 278
Computer Problems 281

6
6.1

Analysis of Structures
Introduction

284

286

Trusses 287
6.2
6.3

6.4
*6.5
*6.6
6.7
*6.8

Definition of a Truss 287
Simple Trusses 289
Analysis of Trusses by the Method of Joints 290
Joints under Special Loading Conditions 292
Space Trusses 294
Analysis of Trusses by the Method of Sections 304
Trusses Made of Several Simple Trusses 305

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Frames and Machines 316

Contents

6.9 Structures Containing Multiforce Members 316
6.10 Analysis of a Frame 316
6.11 Frames Which Cease to Be Rigid When Detached

from Their Supports 317
6.12 Machines 331
Review and Summary 345
Review Problems 348
Computer Problems 350

7
*7.1
*7.2

Forces in Beams and Cables
Introduction 354
Internal Forces in Members

352

354

Beams 362
*7.3
*7.4
*7.5
*7.6

Various Types of Loading and Support 362
Shear and Bending Moment in a Beam 363
Shear and Bending-Moment Diagrams 365
Relations among Load, Shear, and Bending Moment

*7.7

*7.8
*7.9
*7.10

Cables with Concentrated Loads 383
Cables with Distributed Loads 384
Parabolic Cable 385
Catenary 395

Cables

383

Review and Summary 403
Review Problems 406
Computer Problems 408

8

Friction

410

8.1
8.2

Introduction 412
The Laws of Dry Friction. Coefficients
of Friction 412
8.3 Angles of Friction 415

8.4 Problems Involving Dry Friction 416
8.5 Wedges 429
8.6 Square-Threaded Screws 430
*8.7 Journal Bearings. Axle Friction 439
*8.8 Thrust Bearings. Disk Friction 441
*8.9 Wheel Friction. Rolling
Resistance 442
*8.10 Belt Friction 449
Review and Summary 460
Review Problems 463
Computer Problems 467

373


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9
9.1

Distributed Forces:
Moments of Inertia
Introduction

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Contents

470

472

Moments of Inertia of Areas

473

9.2
9.3

Second Moment, or Moment of Inertia, of an Area 473
Determination of the Moment of Inertia of an
Area by Integration 474
9.4 Polar Moment of Inertia 475
9.5 Radius of Gyration of an Area 476
9.6 Parallel-Axis Theorem 483
9.7 Moments of Inertia of Composite Areas 484
*9.8 Product of Inertia 497
*9.9 Principal Axes and Principal Moments of Inertia 498
*9.10 Mohr’s Circle for Moments and Products of Inertia 506

Moments of Inertia of a Mass
9.11
9.12
9.13
9.14
9.15
*9.16
*9.17
*9.18

512

Moment of Inertia of a Mass 512
Parallel-Axis Theorem 514
Moments of Inertia of Thin Plates 515
Determination of the Moment of Inertia of a
Three-Dimensional Body by Integration 516
Moments of Inertia of Composite Bodies 516
Moment of Inertia of a Body with Respect to an Arbitrary Axis
through O. Mass Products of Inertia 532
Ellipsoid of Inertia. Principal Axes of Inertia 533
Determination of the Principal Axes and Principal Moments of
Inertia of a Body of Arbitrary Shape 535

Review and Summary 547
Review Problems 553
Computer Problems 555

10

Method of Virtual Work

*10.1
*10.2
*10.3
*10.4
*10.5
*10.6
*10.7
*10.8

*10.9

Introduction 558
Work of a Force 558
Principle of Virtual Work 561
Applications of the Principle of Virtual Work 562
Real Machines. Mechanical Efficiency 564
Work of a Force during a Finite Displacement 578
Potential Energy 580
Potential Energy and Equilibrium 581
Stability of Equilibrium 582

Review and Summary 592
Review Problems 595
Computer Problems 598

556

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11

11.1

Kinematics of Particles
Introduction to Dynamics

600

602

Rectilinear Motion of Particles 603
11.2
11.3
11.4
11.5
11.6
*11.7
*11.8

Position, Velocity, and Acceleration 603
Determination of the Motion of a Particle 607
Uniform Rectilinear Motion 616
Uniformly Accelerated Rectilinear Motion 617
Motion of Several Particles 618
Graphical Solution of Rectilinear-Motion Problems
Other Graphical Methods 631

630

Curvilinear Motion of Particles 641
11.9

11.10
11.11
11.12
11.13
11.14

Position Vector, Velocity, and Acceleration 641
Derivatives of Vector Functions 643
Rectangular Components of Velocity and Acceleration
Motion Relative to a Frame in Translation 646
Tangential and Normal Components 665
Radial and Transverse Components 668

Review and Summary 682
Review Problems 686
Computer Problems 688

12
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
*12.11
*12.12

*12.13

Kinetics of Particles:
Newton’s Second Law

690

Introduction 692
Newton’s Second Law of Motion 693
Linear Momentum of a Particle. Rate of Change
of Linear Momentum 694
Systems of Units 695
Equations of Motion 697
Dynamic Equilibrium 699
Angular Momentum of a Particle. Rate of Change
of Angular Momentum 721
Equations of Motion in Terms of Radial and
Transverse Components 722
Motion under a Central Force. Conservation of
Angular Momentum 723
Newton’s Law of Gravitation 724
Trajectory of a Particle under a Central Force 734
Application to Space Mechanics 735
Kepler’s Laws of Planetary Motion 738

Review and Summary 746
Review Problems 750
Computer Problems 753

645

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13
13.1
13.2
13.3
13.4
13.5
13.6
*13.7
13.8
13.9
13.10
13.11
13.12
13.13
13.14
13.15

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Kinetics of Particles: Energy and
Momentum Methods 754
Introduction 756
Work of a Force 756
Kinetic Energy of a Particle. Principle of Work
and Energy 760
Applications of the Principle of Work and Energy

Power and Efficiency 763
Potential Energy 782
Conservative Forces 784
Conservation of Energy 785
Motion under a Conservative Central Force.
Application to Space Mechanics 787
Principle of Impulse and Momentum 806
Impulsive Motion 809
Impact 821
Direct Central Impact 821
Oblique Central Impact 824
Problems Involving Energy and Momentum 827

762

Review and Summary 843
Review Problems 849
Computer Problems 852

14

Systems of Particles

14.1
14.2

854

Introduction 856
Application of Newton’s Laws to the Motion of a System

of Particles. Effective Forces 856
14.3 Linear and Angular Momentum of a System of Particles 859
14.4 Motion of the Mass Center of a System of Particles 860
14.5 Angular Momentum of a System of Particles about Its
Mass Center 862
14.6 Conservation of Momentum for a System of Particles 864
14.7 Kinetic Energy of a System of Particles 872
14.8 Work-Energy Principle. Conservation of Energy for a System
of Particles 874
14.9 Principle of Impulse and Momentum for a System
of Particles 874
*14.10 Variable Systems of Particles 885
*14.11 Steady Stream of Particles 885
*14.12 Systems Gaining or Losing Mass 888
Review and Summary 905
Review Problems 909
Computer Problems 912

Contents

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15
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
*15.9
15.10
15.11
*15.12
*15.13
*15.14
*15.15

Kinematics of Rigid Bodies

914

Introduction 916
Translation 918
Rotation about a Fixed Axis 919
Equations Defining the Rotation of a Rigid Body
about a Fixed Axis 922
General Plane Motion 932
Absolute and Relative Velocity in Plane Motion 934
Instantaneous Center of Rotation in Plane Motion 946

Absolute and Relative Acceleration in
Plane Motion 957
Analysis of Plane Motion in Terms of a Parameter 959
Rate of Change of a Vector with Respect to a
Rotating Frame 971
Plane Motion of a Particle Relative to a Rotating Frame.
Coriolis Acceleration 973
Motion about a Fixed Point 984
General Motion 987
Three-Dimensional Motion of a Particle Relative to a Rotating
Frame. Coriolis Acceleration 998
Frame of Reference in General Motion 999

Review and Summary 1011
Review Problems 1018
Computer Problems 1021

16
16.1
16.2
16.3
16.4
*16.5
16.6
16.7
16.8

Plane Motion of Rigid Bodies: Forces
and Accelerations 1024
Introduction 1026

Equations of Motion for a Rigid Body 1027
Angular Momentum of a Rigid Body in
Plane Motion 1028
Plane Motion of a Rigid Body.
D’Alembert’s Principle 1029
A Remark on the Axioms of the Mechanics
of Rigid Bodies 1030
Solution of Problems Involving the Motion of a
Rigid Body 1031
Systems of Rigid Bodies 1032
Constrained Plane Motion 1052

Review and Summary 1074
Review Problems 1076
Computer Problems 1079


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17

Plane Motion of Rigid Bodies: Energy
and Momentum Methods 1080

17.1
17.2
17.3
17.4
17.5
17.6

17.7
17.8

Introduction 1082
Principle of Work and Energy for a Rigid Body 1082
Work of Forces Acting on a Rigid Body 1083
Kinetic Energy of a Rigid Body in Plane Motion 1084
Systems of Rigid Bodies 1085
Conservation of Energy 1086
Power 1087
Principle of Impulse and Momentum for the Plane Motion
of a Rigid Body 1103
17.9 Systems of Rigid Bodies 1106
17.10 Conservation of Angular Momentum 1106
17.11 Impulsive Motion 1119
17.12 Eccentric Impact 1119

Review and Summary 1135
Review Problems 1139
Computer Problems 1142

18

Kinetics of Rigid Bodies in
Three Dimensions 1144

*18.1
*18.2

Introduction 1146

Angular Momentum of a Rigid Body in
Three Dimensions 1147
*18.3 Application of the Principle of Impulse and Momentum to the
Three-Dimensional Motion of a Rigid Body 1151
*18.4 Kinetic Energy of a Rigid Body in
Three Dimensions 1152
*18.5 Motion of a Rigid Body in Three Dimensions 1165
*18.6 Euler’s Equations of Motion. Extension of
D’Alembert’s Principle to the Motion of a
Rigid Body in Three Dimensions 1166
*18.7 Motion of a Rigid Body about a Fixed Point 1167
*18.8 Rotation of a Rigid Body about a Fixed Axis 1168
*18.9 Motion of a Gyroscope. Eulerian Angles 1184
*18.10 Steady Precession of a Gyroscope 1186
*18.11 Motion of an Axisymmetrical Body
under No Force 1187
Review and Summary 1201
Review Problems 1206
Computer Problems 1209

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19
19.1

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Mechanical Vibrations
Introduction

1214

Vibrations without Damping
19.2
19.3
*19.4
19.5
19.6
19.7

1212

1214

Free Vibrations of Particles. Simple Harmonic Motion 1214
Simple Pendulum (Approximate Solution) 1218
Simple Pendulum (Exact Solution) 1219
Free Vibrations of Rigid Bodies 1228
Application of the Principle of Conservation of Energy 1240

Forced Vibrations 1250

Damped Vibrations 1260
*19.8 Damped Free Vibrations 1260
*19.9 Damped Forced Vibrations 1263
*19.10 Electrical Analogues 1264
Review and Summary 1277
Review Problems 1282
Computer Problems 1286

Appendix Fundamentals of Engineering Examination
Photo Credits 1291
Index 1293
Answers to Problems 1305

1289


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Preface
OBJECTIVES
The main objective of a first course in mechanics should be to
develop in the engineering student the ability to analyze any problem
in a simple and logical manner and to apply to its solution a few, wellunderstood, basic principles. This text is designed for the first courses
in statics and dynamics offered in the sophomore or junior year, and
it is hoped that it will help the instructor achieve this goal.†

GENERAL APPROACH
Vector analysis is introduced early in the text and is used throughout
the presentation of statics and dynamics. This approach leads to more
concise derivations of the fundamental principles of mechanics. It also
results in simpler solutions of three-dimensional problems in statics
and makes it possible to analyze many advanced problems in kinematics and kinetics, which could not be solved by scalar methods. The
emphasis in this text, however, remains on the correct understanding
of the principles of mechanics and on their application to the solution
of engineering problems, and vector analysis is presented chiefly as a
convenient tool.‡

Practical Applications Are Introduced Early. One of the characteristics of the approach used in this book is that mechanics of
particles is clearly separated from the mechanics of rigid bodies. This
approach makes it possible to consider simple practical applications
at an early stage and to postpone the introduction of the more difficult concepts. For example:
• In Statics, the statics of particles is treated first (Chap. 2); after
the rules of addition and subtraction of vectors are introduced,
the principle of equilibrium of a particle is immediately applied
to practical situations involving only concurrent forces. The statics of rigid bodies is considered in Chaps. 3 and 4. In Chap. 3,
the vector and scalar products of two vectors are introduced and
used to define the moment of a force about a point and about
an axis. The presentation of these new concepts is followed by a
thorough and rigorous discussion of equivalent systems of forces
leading, in Chap. 4, to many practical applications involving the
equilibrium of rigid bodies under general force systems.

†This text is available in separate volumes, Vector Mechanics for Engineers: Statics, ninth
edition, and Vector Mechanics for Engineers: Dynamics, ninth edition.
‡In a parallel text, Mechanics for Engineers: fifth edition, the use of vector algebra is
limited to the addition and subtraction of vectors, and vector differentiation is omitted.

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• In Dynamics, the same division is observed. The basic concepts of
force, mass, and acceleration, of work and energy, and of impulse
and momentum are introduced and first applied to problems involving only particles. Thus, students can familiarize themselves
with the three basic methods used in dynamics and learn their
respective advantages before facing the difficulties associated
with the motion of rigid bodies.

New Concepts Are Introduced in Simple Terms. Since this text
is designed for the first course in statics and dynamics, new concepts
are presented in simple terms and every step is explained in detail.
On the other hand, by discussing the broader aspects of the problems considered, and by stressing methods of general applicability, a
definite maturity of approach is achieved. For example:
• In Statics, the concepts of partial constraints and statical indeterminacy are introduced early and are used throughout statics.
• In Dynamics, the concept of potential energy is discussed in the
general case of a conservative force. Also, the study of the plane
motion of rigid bodies is designed to lead naturally to the study
of their general motion in space. This is true in kinematics as well
as in kinetics, where the principle of equivalence of external and

effective forces is applied directly to the analysis of plane motion,
thus facilitating the transition to the study of three-dimensional
motion.

Fundamental Principles Are Placed in the Context of Simple
Applications. The fact that mechanics is essentially a deductive
science based on a few fundamental principles is stressed. Derivations
have been presented in their logical sequence and with all the rigor
warranted at this level. However, the learning process being largely
inductive, simple applications are considered first. For example:
• The statics of particles precedes the statics of rigid bodies, and
problems involving internal forces are postponed until Chap. 6.
• In Chap. 4, equilibrium problems involving only coplanar forces
are considered first and solved by ordinary algebra, while problems involving three-dimensional forces and requiring the full use
of vector algebra are discussed in the second part of the chapter.
• The kinematics of particles (Chap. 11) precedes the kinematics
of rigid bodies (Chap. 15).
• The fundamental principles of the kinetics rigid bodies are first
applied to the solution of two-dimensional problems (Chaps. 16
and 17), which can be more easily visualized by the student, while
three-dimensional problems are postponed until Chap. 18.

The Presentation of the Principles of Kinetics Is Unified. The
ninth edition of Vector Mechanics for Engineers retains the unified
presentation of the principles of kinetics which characterized the previous eight editions. The concepts of linear and angular momentum are
introduced in Chap. 12, so that Newton’s second law of motion can be
presented not only in its conventional form F 5 ma, but also as a law
relating, respectively, the sum of the forces acting on a particle and the

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sum of their moments to the rates of change of the linear and angular
momentum of the particle. This makes possible an earlier introduction
of the principle of conservation of angular momentum and a more
meaningful discussion of the motion of a particle under a central force
(Sec. 12.9). More importantly, this approach can be readily extended
to the study of the motion of a system of particles (Chap. 14) and leads
to a more concise and unified treatment of the kinetics of rigid bodies
in two and three dimensions (Chaps. 16 through 18).

Free-Body Diagrams Are Used Both to Solve Equilibrium
Problems and to Express the Equivalence of Force Systems.
Free-body diagrams are introduced early, and their importance is
emphasized throughout the text. They are used not only to solve
equilibrium problems but also to express the equivalence of two systems of forces or, more generally, of two systems of vectors. The
advantage of this approach becomes apparent in the study of the
dynamics of rigid bodies, where it is used to solve three-dimensional
as well as two-dimensional problems. By placing the emphasis on
“free-body-diagram equations” rather than on the standard algebraic
equations of motion, a more intuitive and more complete understanding of the fundamental principles of dynamics can be achieved.
This approach, which was first introduced in 1962 in the first edition
of Vector Mechanics for Engineers, has now gained wide acceptance
among mechanics teachers in this country. It is, therefore, used in
preference to the method of dynamic equilibrium and to the equations of motion in the solution of all sample problems in this book.
A Four-Color Presentation Uses Color to Distinguish Vectors.
Color has been used, not only to enhance the quality of the illustrations,
but also to help students distinguish among the various types of vectors they will encounter. While there is no intention to “color code”
this text, the same color is used in any given chapter to represent vectors of the same type. Throughout Statics, for example, red is used
exclusively to represent forces and couples, while position vectors are

shown in blue and dimensions in black. This makes it easier for the
students to identify the forces acting on a given particle or rigid body
and to follow the discussion of sample problems and other examples
given in the text. In Dynamics, for the chapters on kinetics, red is used
again for forces and couples, as well as for effective forces. Red is also
used to represent impulses and momenta in free-body-diagram equations, while green is used for velocities, and blue for accelerations. In
the two chapters on kinematics, which do not involve any forces, blue,
green, and red are used, respectively, for displacements, velocities, and
accelerations.
A Careful Balance Between SI and U.S. Customary Units Is
Consistently Maintained. Because of the current trend in the
American government and industry to adopt the international system of units (SI metric units), the SI units most frequently used in
mechanics are introduced in Chap. 1 and are used throughout the
text. Approximately half of the sample problems and 60 percent of
the homework problems are stated in these units, while the remainder

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are in U.S. customary units. The authors believe that this approach
will best serve the need of students, who, as engineers, will have to
be conversant with both systems of units.
It also should be recognized that using both SI and U.S. customary units entails more than the use of conversion factors. Since the SI
system of units is an absolute system based on the units of time, length,
and mass, whereas the U.S. customary system is a gravitational system
based on the units of time, length, and force, different approaches are
required for the solution of many problems. For example, when SI
units are used, a body is generally specified by its mass expressed in
kilograms; in most problems of statics it will be necessary to determine
the weight of the body in newtons, and an additional calculation will
be required for this purpose. On the other hand, when U.S. customary
units are used, a body is specified by its weight in pounds and, in
dynamics problems, an additional calculation will be required to determine its mass in slugs (or lb ? s2/ft). The authors, therefore, believe
that problem assignments should include both systems of units.
The Instructor’s and Solutions Manual provides six different
lists of assignments so that an equal number of problems stated in
SI units and in U.S. customary units can be selected. If so desired,
two complete lists of assignments can also be selected with up to
75 percent of the problems stated in SI units.

Optional Sections Offer Advanced or Specialty Topics. A
large number of optional sections have been included. These sections
are indicated by asterisks and thus are easily distinguished from those
which form the core of the basic mechanics course. They may be omitted without prejudice to the understanding of the rest of the text.
The topics covered in the optional sections in statics include
the reduction of a system of forces to a wrench, applications to hydrostatics, shear and bending-moment diagrams for beams, equilibrium
of cables, products of inertia and Mohr’s circle, mass products of
inertia and principal axes of inertia for three-dimensional bodies, and

the method of virtual work. An optional section on the determination
of the principal axes and the mass moments of inertia of a body of
arbitrary shape is included (Sec. 9.18). The sections on beams are
especially useful when the course in statics is immediately followed
by a course in mechanics of materials, while the sections on the inertia
properties of three-dimensional bodies are primarily intended for the
students who will later study in dynamics the three-dimensional motion
of rigid bodies.
The topics covered in the optional sections in dynamics
include graphical methods for the solution of rectilinear-motion
problems, the trajectory of a particle under a central force, the
deflection of fluid streams, problems involving jet and rocket propulsion, the kinematics and kinetics of rigid bodies in three dimensions, damped mechanical vibrations, and electrical analogues.
These topics will be found of particular interest when dynamics is
taught in the junior year.
The material presented in the text and most of the problems
require no previous mathematical knowledge beyond algebra, trigonometry, and elementary calculus; all the elements of vector algebra


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necessary to the understanding of the text are carefully presented in
Chaps. 2 and 3. However, special problems are included, which make
use of a more advanced knowledge of calculus, and certain sections,
such as Secs. 19.8 and 19.9 on damped vibrations, should be assigned
only if students possess the proper mathematical background. In portions of the text using elementary calculus, a greater emphasis is
placed on the correct understanding and application of the concepts
of differentiation and integration than on the nimble manipulation
of mathematical formulas. In this connection, it should be mentioned
that the determination of the centroids of composite areas precedes
the calculation of centroids by integration, thus making it possible to

establish the concept of moment of area firmly before introducing
the use of integration.

CHAPTER ORGANIZATION AND PEDAGOGICAL FEATURES
Chapter Introduction. Each chapter begins with an introductory
section setting the purpose and goals of the chapter and describing
in simple terms the material to be covered and its application to the
solution of engineering problems. Chapter outlines provide students
with a preview of chapter topics.
Chapter Lessons. The body of the text is divided into units, each
consisting of one or several theory sections, one or several sample
problems, and a large number of problems to be assigned. Each unit
corresponds to a well-defined topic and generally can be covered in
one lesson. In a number of cases, however, the instructor will find it
desirable to devote more than one lesson to a given topic. The
Instructor’s and Solutions Manual contains suggestions on the coverage of each lesson.
Sample Problems. The sample problems are set up in much the
same form that students will use when solving the assigned problems.
They thus serve the double purpose of amplifying the text and demonstrating the type of neat, orderly work that students should cultivate in their own solutions.
Solving Problems on Your Own. A section entitled Solving
Problems on Your Own is included for each lesson, between the
sample problems and the problems to be assigned. The purpose of
these sections is to help students organize in their own minds the
preceding theory of the text and the solution methods of the sample
problems so that they can more successfully solve the homework
problems. Also included in these sections are specific suggestions
and strategies which will enable students to more efficiently attack
any assigned problems.
Homework Problem Sets. Most of the problems are of a practical nature and should appeal to engineering students. They are primarily designed, however, to illustrate the material presented in the

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text and to help students understand the principles of mechanics.
The problems are grouped according to the portions of material they
illustrate and are arranged in order of increasing difficulty. Problems
requiring special attention are indicated by asterisks. Answers to
70 percent of the problems are given at the end of the book. Problems
for which the answers are given are set in straight type in the text,
while problems for which no answer is given are set in italic.

Chapter Review and Summary. Each chapter ends with a
review and summary of the material covered in that chapter. Marginal notes are used to help students organize their review work, and
cross-references have been included to help them find the portions
of material requiring their special attention.
Review Problems. A set of review problems is included at the end
of each chapter. These problems provide students further opportunity
to apply the most important concepts introduced in the chapter.

Computer Problems. Each chapter includes a set of problems
designed to be solved with computational software. Many of these
problems provide an introduction to the design process. In Statics,
for example, they may involve the analysis of a structure for various
configurations and loading of the structure or the determination of
the equilibrium positions of a mechanism which may require an iterative method of solution. In Dynamics, they may involve the determination of the motion of a particle under initial conditions, the kinematic
or kinetic analysis of mechanisms in successive positions, or the
numerical integration of various equations of motion. Developing the
algorithm required to solve a given mechanics problem will benefit
the students in two different ways: (1) it will help them gain a better
understanding of the mechanics principles involved; (2) it will provide
them with an opportunity to apply their computer skills to the solution of a meaningful engineering problem.

SUPPLEMENTS
An extensive supplements package for both instructors and students
is available with the text.

Instructor’s and Solutions Manual. The Instructor’s and Solutions
Manual that accompanies the ninth edition features typeset, one-perpage solutions to all homework problems. This manual also features
a number of tables designed to assist instructors in creating a schedule of assignments for their courses. The various topics covered in the
text are listed in Table I, and a suggested number of periods to be
spent on each topic is indicated. Table II provides a brief description
of all groups of problems and a classification of the problems in each
group according to the units used. Sample lesson schedules are
shown in Tables III, IV, and V.


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McGRAW-HILL CONNECT ENGINEERING

McGraw-Hill Connect Engineering is a web-based assignment and
assessment platform that gives students the means to better connect
with their coursework, their instructors, and the important concepts
that they will need to know for success now and in the future. With
Connect Engineering, instructors can deliver assignments, quizzes,
and tests easily online. Students can practice important skills at their
own pace and on their own schedule.
Connect Engineering for Vector Mechanics for Engineers is
available at www.mhhe.com/beerjohnston and includes algorithmic
problems from the text, Lecture PowerPoints, an image bank, and
animations.

Hands-on Mechanics. Hands-on Mechanics is a website designed
for instructors who are interested in incorporating three-dimensional,
hands-on teaching aids into their lectures. Developed through a
partnership between the McGraw-Hill Engineering Team and the
Department of Civil and Mechanical Engineering at the United
States Military Academy at West Point, this website not only provides detailed instructions for how to build 3-D teaching tools using
materials found in any lab or local hardware store but also provides
a community where educators can share ideas, trade best practices,
and submit their own demonstrations for posting on the site. Visit
www.handsonmechanics.com.

ELECTRONIC TEXTBOOK OPTIONS
Ebooks are an innovative way for students to save money and create
a greener environment at the same time. An ebook can save students
about half the cost of a traditional textbook and offers unique features
like a powerful search engine, highlighting, and the ability to share
notes with classmates using ebooks.
McGraw-Hill offers two ebook options: purchasing a downloadable book from VitalSource or a subscription to the book from CourseSmart. To talk about the ebook options, contact your McGraw-Hill

sales rep or visit the sites directly at www.vitalsource.com and
www.coursesmart.com.

ACKNOWLEDGMENTS
A special thanks go to our colleagues who thoroughly checked the
solutions and answers of all problems in this edition and then prepared the solutions for the accompanying Instructor’s and Solution
Manual: Amy Mazurek of Williams Memorial Institute and Dean
Updike of Lehigh University.
We are pleased to recognize Dennis Ormond of Fine Line
Illustrations for the artful illustrations which contribute so much to
the effectiveness of the text.
The authors thank the many companies that provided photographs for this edition. We also wish to recognize the determined
efforts and patience of our photo researcher Sabina Dowell.

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The authors gratefully acknowledge the many helpful comments and suggestions offered by users of the previous editions of

Vector Mechanics for Engineers.
E. Russell Johnston, Jr.
David Mazurek
Phillip Cornwell
Elliot R. Eisenberg


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List of Symbols
a
A, B, C, . . .
A, B, C, . . .
A
b
c
C
d
e
F
g
G
h
i, j, k
I, Ix, . . .
I
Ixy, . . .
J

k
kx, ky, kO
k
l
L
m
M
MO
MOR
M
MOL
N
O
p
P
Q
r
r
R
R
s
s
S
t
T
T
U

Constant; radius; distance
Reactions at supports and connections

Points
Area
Width; distance
Constant
Centroid
Distance
Base of natural logarithms
Force; friction force
Acceleration of gravity
Center of gravity; constant of gravitation
Height; sag of cable
Unit vectors along coordinate axes
Moments of inertia
Centroidal moment of inertia
Products of inertia
Polar moment of inertia
Spring constant
Radii of gyration
Centroidal radius of gyration
Length
Length; span
Mass
Couple; moment
Moment about point O
Moment resultant about point O
Magnitude of couple or moment; mass of earth
Moment about axis OL
Normal component of reaction
Origin of coordinates
Pressure

Force; vector
Force; vector
Position vector
Radius; distance; polar coordinate
Resultant force; resultant vector; reaction
Radius of earth
Position vector
Length of arc; length of cable
Force; vector
Thickness
Force
Tension
Work

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V
V
w
W, W
x, y, z

x, y, z
a, b, g
g
d
dr
dU
L
h
u
m
r
f

Vector product; shearing force
Volume; potential energy; shear
Load per unit length
Weight; load
Rectangular coordinates; distances
Rectangular coordinates of centroid or center of
gravity
Angles
Specific weight
Elongation
Virtual displacement
Virtual work
Unit vector along a line
Efficiency
Angular coordinate; angle; polar coordinate
Coefficient of friction
Density

Angle of friction; angle